Dependencies

We define \(a := -\sum _m \nu (m) \log m / m\).

We have \(0.92129 \leq a \leq 0.92130\).

Let \(M{\gt}0\). Let \(f\) be analytic on the closed ball \(|z|\leq R\) such that \(f(0)=0\) and suppose that \(2M - f(z)\neq 0\) for all \(|z|\leq R\). Then, with \(f_{M}\) defined as in Definition 4.3.2, \(f_{M}\) is analytic on \(|z|\leq R\).

Let \(f\) be a complex function analytic on an open set \(s\) containing \(0\) such that \(f(0)=0\). Then, with \(g\) defined as in Definition 4.3.1, \(g\) is analytic on \(s\).

Let \(f\) be a complex function analytic on the closed ball \(|z|\leq R\) such that \(f(0)=0\). Then, with \(g\) defined as in Definition 4.3.1, \(g\) is analytic on \(|z|\leq R\).

One has

for all \(x \geq 1\).

The balance of a factorization \(a_1 \dots a_t\) at a prime \(p\) is defined as the number of times \(p\) divides \(a_1 \dots a_t\), minus the number of times \(p\) divides \(n!\).

If a factorization has zero total imbalance, then it exactly factors \(n!\).

If \(f\) is bounded above in a punctured neighborhood of \(p\), then \(f\) is \(O(1)\) in that neighborhood.

Let \(g : \mathbb {C}\to \mathbb {C}\) be a holomorphic function on a rectangle, then \(g\) is bounded above on the rectangle.

The Fourier transform is a bijection on the Schwartz class. [Note: only surjectivity is actually used.]

Suppose one has an asymptotic bound \(E_\psi \) with parameters \(A,B,C,R,e^{x_0}\) (which need to satisfy some additional bounds) with \(x_0 \geq 1000\). Then \(E_\theta \) obeys an asymptotic bound with parameters \(A', B, C, R, e^{x_0}\), where

and \(a_1(x_0), a_2(x_0)\) are as in Corollary 9.7.3.

\(\theta (x) \leq (1 + 1.93378 \times 10^{-8}) x\).

Let \(b \geq 7\). Assume \(x \geq e^b\). Then we have

where

Let \(b \geq 7\). Then for all \(x \geq e^b\) we have \(\psi (x) - \vartheta (x) {\lt} a_1 x^{1/2} + a_2 x^{1/3}\), where \(a_1 = a_1(b) = 1 + 1.93378 \times 10^{-8}\) if \(b \leq 38 \log 10\), \(1 + \varepsilon (b/2)\) if \(b {\gt} 38 \log 10\), and \(a_2 = a_2(b) = (1 + 1.93378 \times 10^{-8}) \max \left( f(e^b), f(2^{\lfloor \frac{b}{\log 2} \rfloor + 1}) \right)\), where \(f\) is defined by (2.4) and values for \(\varepsilon (b/2)\) are from Table 8.

Let \(b\) be a positive constant such that \(\log 11 {\lt} b \leq 19 \log (10)\). Then we have

Note that by Table 8, we have \(\varepsilon (19 \log 10) = 1.93378 \cdot 10^{-8}\).

Let \(b\) be a positive constant such that \(\log 11 {\lt} b \leq 19 \log (10)\). Then we have

\(a_1\) is equal to \(1 + \varepsilon (\log x_1)\) if \(b \leq 2\log x_1\), and equal to \(1 + \varepsilon (b/2)\) if \(b {\gt} 2\log x_1\).

\(a_2\) is defined by

We have

We denote

We have

We have

We denote

We define

We define

We define

We define

We define

We define

and

For \(100 \leq x \leq 10^{19}\), one has

Let \(x \geq e^{1000}\) and \(T\) satisfies \(50 {\lt} T \leq x\). Then

where \(A = \mathcal{O}^*(B)\) means \(|A| \leq B\).

We denote

We have

We take \(\alpha = 1.93378 \times 10^{-8}\), so that we have \(\theta (x) \leq (1 + \alpha ) x\) for all \(x\).

Suppose there exists \(c_1, c_2, c_3, c_4 {\gt} 0\) such that

Let \(k {\gt} 0\) and let \(b \geq \max \left(\log c_4, \log \left(\frac{4(c_2 + k)^2}{c_3^2}\right)\right)\). Then for all \(x \geq e^b\) we have

where

With the hypotheses as above, we have \(\theta (x) \leq (1+\varepsilon (\log x_1)) x)\) for all \(x {\gt} 0\).

With the hypotheses as above, we have

for all \(x \geq e^b\) and \(b{\gt}0\), where \(c_0 = 1.03883\) is the constant from [ 20 , Theorem 12 ] .

Let \(B_0\), \(B\), and \(c\) be positive constants such that

is known. Furthermore, assume for every \(b_0 {\gt} 0\) there exists \(\varepsilon (b_0) {\gt} 0\) such that

Let \(b\) be positive such that \(e^b \in (B_0, B]\). Then, for all \(x \geq e^b\) we have

We take \(\varepsilon \) as in Table 8 of [ 2 ] , and \(x_1 = 10^{19}\).

Let \(x_0 \geq 2^9\). Let \(\alpha {\gt} 0\) exist such that \(\theta (x) \leq (1 + \alpha )x\) for \(x {\gt} 0\). Then for \(x \geq x_0\),

where

with

Let \(x \geq x_0\) and let \(\alpha \) be admissible. Then

\(f\) decreases on \([2^n, 2^{n+1})\).

\(f(2^n) = 1 + u_n\).

\(u_{n+1} {\lt} u_n\) for \(n \geq 9\).

\(f(2^n) {\gt} f(2^{n+1})\) for \(n \geq 9\).

\(f(x) \leq f(2^{\lfloor \frac{\log x_0}{\log 2} \rfloor + 1})\) on \([2^{\lfloor \frac{\log x_0}{\log 2} \rfloor + 1}, \infty )\).

\(f(x) \leq f(x_0)\) for \(x \in [x_0, 2^{\lfloor \frac{\log x_0}{\log 2} \rfloor + 1})\).

\(f(x) \leq \max \left(f(x_0), f(2^{\lfloor \frac{\log x_0}{\log 2} \rfloor + 1})\right)\).

If \(7 \leq b \leq 2\log x_1\), then we have

If \(b {\gt} 2\log x_1\), then we have

We denote

We denote

The entries in Table 14 obey the criterion in Sublemma 9.7.2.

The value of \(\varepsilon (b)\) arising from Table 8 of [ 2 ] is weaker than that from the expanded version of Table 8 available in the arXiv.

Let \(b_1, b_2\) satisfy \(1000 \leq b_1 {\lt} b_2\). Let \(0.001 \leq \delta \leq 0.025\), \(\lambda {\gt} 1\), \(H {\lt} T {\lt} e^{b_1}\), and \(K = \left\lfloor \frac{\log \frac{T}{H}}{\log \lambda } \right\rfloor + 1\). Then for all \(x \in [e^{b_1}, e^{b_2}]\)

where \(s_0, s_1, s_2\) are respectively defined in Definitions 8.5.1, 8.5.4, and 8.5.5

Let \(x_0 \geq 1000\) and let \(\sigma \in [0.75, 1)\). For all \(x \geq e^{x_0}\),

where \(A\), \(B\), and \(C\) are defined in Definitions 8.5.10, 8.5.11.

For any fixed \(X_0 \geq 1\), there exists \(m_0 {\gt} 0\) such that, for all \(x \geq X_0\)

For any fixed \(X_1 \geq 1\), there exists \(M_0 {\gt} 0\) such that, for all \(x \geq X_1\)

For \(X_0, X_1 \geq e^{20}\), we have

and

One has \(x(1-m) \leq \theta (x) \leq x(1+M)\) whenever \(x \geq e^b\) and \(b,M,m\) obey the condition that \(b \geq 20\), \(\varepsilon (b) \leq M\), and \(\varepsilon (b) + c_0 (e^{-b/2} + e^{-2b/3} + e^{-4b/5}) \leq m\).

The previous theorem holds with \((b,M,m)\) given by the values in [ 2 , Table 14 ] .

Let \(k\) be an integer with \(1 \leq k \leq 5\). For any fixed \(X_0 {\gt} 1\), there exists \(m_k {\gt} 0\) such that, for all \(x \geq X_0\)

For any fixed \(X_1 {\gt} 1\), there exists \(M_k {\gt} 0\) such that, for all \(x \geq X_1\)

In the case \(k = 0\) and \(X_0, X_1 \geq e^{20}\), we have

and

See [ 2 , Table 15 ] for values of \(m_k\) and \(M_k\), for \(k \in \{ 1,2,3,4,5\} \).

Let \(\alpha {\gt} 0\) exist such that

Assume for every \(b \geq 7\) there exists a positive constant \(\varepsilon (b)\) such that

Assume there exists \(x_1 \geq e^7\) such that

Let \(b \geq 7\). Then, for all \(x \geq e^b\) we have

where

and

Let \(0 {\lt} \varepsilon {\lt} 10^{-3}\), \(c \geq 3\), \(x_0 \geq 100\) and \(\alpha \in [0, 1)\) such that the inequality

holds. We denote the zeros of the Riemann zeta function by \(\rho = \beta + i\gamma \) with \(\beta , \gamma \in \mathbb {R}\). Then, if \(\beta = \frac{1}{2}\) holds for \(0 {\lt} \gamma \leq \frac{c}{\varepsilon }\), the inequality

holds for all \(x \geq e^{\varepsilon \alpha } x_0\), where

The \(\nu _c(\alpha ) = \nu _{c,1}(\alpha )\) and \(\mu _c(\alpha ) = \mu _{c,1}(\alpha )\) where \(\nu _{c,\varepsilon }(\alpha )\) and \(\mu _{c,\varepsilon }(\alpha )\) are defined by [ 3 , p. 2490 ] .

If \(b{\gt}0\) then \(|\psi (x) - x| \leq \varepsilon (b) x\) for all \(x \geq \exp (b)\), where \(\varepsilon \) is as in [ 2 , Table 8 ] .

Let \(0 {\lt} r {\lt} R{\lt}1\), and \(f:\overline{\mathbb {D}_1}\to \mathbb {C}\) be analytic on neighborhoods of points in \(\overline{\mathbb {D}_1}\) with \(f(0)\neq 0\). We define a function \(B_f:\overline{\mathbb {D}_R}\to \mathbb {C}\) as follows.

Let \(0 {\lt} r {\lt} R{\lt}1\) and \(f:\overline{\mathbb {D}_1}\to \mathbb {C}\) be analytic on neighborhoods of points in \(\overline{\mathbb {D}_1}\). Then \(B_f(z)\neq 0\) for all \(z\in \overline{\mathbb {D}_r}\).

Let \(0 {\lt} r {\lt} R{\lt}1\), and \(f:\overline{\mathbb {D}_1}\to \mathbb {C}\) be analytic on neighborhoods of points in \(\overline{\mathbb {D}_1}\) with \(f(0)\neq 0\). Then

Let \(R,\, M{\gt}0\). Let \(f\) be analytic on \(|z|\leq R\) such that \(f(0)=0\) and suppose \(\Re f(z)\leq M\) for all \(|z|\leq R\). Then for any \(0 {\lt} r {\lt} R\),

Let \(R,\, M{\gt}0\). Let \(f\) be analytic on \(|z|\leq R\) such that \(f(0)=0\) and suppose \(\Re f(z)\leq M\) for all \(|z|\leq R\). Then for any \(0 {\lt} r {\lt} R\),

for all \(|z|\leq r\).

If \(M\) is sufficiently large depending on \(\varepsilon \), then \(n \log (1-1/M)^{-1} \leq \varepsilon n\).

If \(L\) is sufficiently large depending on \(M, \varepsilon \), and \(n\) sufficiently large depending on \(L\), then \(\sum _{p \leq n/L} M \log n \leq \varepsilon n\).

If \(n\) sufficiently large depending on \(M, \varepsilon \), then \(\sum _{p \leq \sqrt{n}} M \log ^2 n / \log 2 \leq \varepsilon n\).

If \(n\) sufficiently large depending on \(L, \varepsilon \), then \(\sum _{n/L {\lt} p \leq n} \frac{n}{p} \log \frac{n}{p} \leq \varepsilon n\).

If \(n\) sufficiently large depending on \(M, L, \varepsilon \), then \(\sum _{p \leq L} (M \log n + M L \pi (n)) \log L \leq \varepsilon n\).

\(\pi ^*(x) = \sum _{k \geq 1} \pi (x^{1/k}) / k\).

We define the auxiliary functions

If \(11 {\lt} x \leq 10^{19}\), then \(|x - \psi (x)| \leq 0.94\sqrt{x}\).

If \(1423 \leq x \leq 10^{19}\), then \(x - \vartheta (x) \leq 1.95\sqrt{x}\).

If \(1 \leq x \leq 10^{19}\), then \(x - \vartheta (x) {\gt} 0.05\sqrt{x}\).

If \(2 \leq x \leq 10^{19}\), then \(|\operatorname {li}(x) - \pi ^*(x)| {\lt} \frac{\sqrt{x}}{\log x}\).

If \(2 \leq x \leq 10^{19}\), then

If \(2 \leq x \leq 10^{19}\), then \(\mathrm{li}(x) - \pi (x) {\gt} 0\).

Starting from any factorization \(f\), one can find a factorization \(f'\) in balance whose cardinality is at most \(\log n!\) plus the score of \(f\), divided by \(\log n\).

Let \(f\) be analytic on \(|z|\leq R\). For any \(z\) with \(|z|\leq r\) and any \(r'\) with \(0 {\lt} r {\lt} r' {\lt} R\) we have

Let \(0 {\lt} r {\lt} R{\lt}1\), and \(f:\overline{\mathbb {D}_1}\to \mathbb {C}\) be analytic on neighborhoods of points in \(\overline{\mathbb {D}_1}\) with \(f(0)\neq 0\). We define a function \(C_f:\overline{\mathbb {D}_R}\to \mathbb {C}\) as follows. This function is constructed by dividing \(f(z)\) by a polynomial whose roots are the zeros of \(f\) inside \(\overline{\mathbb {D}_r}\).

where \(h_z(z)\) comes from Lemma 4.3.9.

\(S_\sigma (x) = x^{-\sigma } \sum _n a_n \frac{x}{n} I_\lambda ( \frac{T}{2\pi } \log \frac{n}{x} )\) where \(\lambda = 2\pi (\sigma -1)/T\).

\(I_\lambda (u) = 1_{[0,\infty )}(\mathrm{sgn}(\lambda )u) e^{-\lambda u}\).

Let \(a_n\) be a sequence with \(\sum _{n{\gt}1} \frac{|a_n|}{n \log ^\beta n} {\lt} \infty \) for some \(\beta {\gt} 1\). Assume that \(\sum _n a_n n^{-s} - \frac{1}{s-1}\) extends continuously to a function \(G\) defined on \(1 + i[-T,T]\). Let \(\varphi \) be absolutely integrable, supported on \([-1,1]\), and has Fourier decay \(\hat\varphi (y) = O(1/|y|^\beta )\). Then for any \(x{\gt}0\),

Let \(a_n\) be a sequence with \(\sum _{n{\gt}1} \frac{|a_n|}{n \log ^\beta n} {\lt} \infty \) for some \(\beta {\gt} 1\). Write \(G(s)= \sum _n a_n n^{-s} - \frac{1}{s-1}\) for \(\mathrm{Re} s {\gt} 1\). Let \(\varphi \) be absolutely integrable, supported on \([-1,1]\), and has Fourier decay \(\hat\psi (y) = O(1/|y|^\beta )\). Then for any \(x{\gt}0\) and \(\sigma {\gt} 1\)

Let \(a_n\) be a non-negative sequence with \(\sum _{n{\gt}1} \frac{|a_n|}{n \log ^\beta n} {\lt} \infty \) for some \(\beta {\gt} 1\). Assume that \(\sum _n a_n n^{-s} - \frac{1}{s-1}\) extends continuously to a function \(G\) defined on \(1 + i[-T,T]\). Let \(\varphi _-\) be absolutely integrable, supported on \([-1,1]\), and has Fourier decay \(\hat\varphi _-(y) = O(1/|y|^\beta )\). Assume \(\hat\varphi _-(y) \leq I_\lambda (y)\) for all \(y\). Then for any \(x\geq 1\) and \(\sigma \neq 1\),

Let \(a_n\) be a non-negative sequence with \(\sum _{n{\gt}1} \frac{|a_n|}{n \log ^\beta n} {\lt} \infty \) for some \(\beta {\gt} 1\). Assume that \(\sum _n a_n n^{-s} - \frac{1}{s-1}\) extends continuously to a function \(G\) defined on \(1 + i[-T,T]\). Let \(\varphi _+\) be absolutely integrable, supported on \([-1,1]\), and has Fourier decay \(\hat\varphi _+(y) = O(1/|y|^\beta )\). Assume \(I_\lambda (y) \leq \hat\varphi _+(y)\) for all \(y\). Then for any \(x\geq 1\) and \(\sigma \neq 1\),

\(S_\sigma (x)\) is equal to \(\sum _{n \leq x} a_n / n^\sigma \) if \(\sigma {\lt} 1\) and \(\sum _{n \geq x} a_n / n^\sigma \) if \(\sigma {\gt} 1\).

For any \(a\) coprime to \(m\),

\(T(x) := \sum _{n \leq x} \log n\).

If \(\nu : \mathbb {N}\to \mathbb {R}\), let \(E: \mathbb {R}\to \mathbb {R}\) denote the function \(E(x):= \sum _m \nu (m) \lfloor x / m \rfloor \).

One has \(E(x)=1\) for \(1 \leq x {\lt} 6\).

One has \(E(x+30) = E(x)\).

One has \(0 \leq E(x) \leq 1\) for all \(x \geq 0\).

\(\nu = e_1 - e_2 - e_3 - e_5 + e_{30}\), where \(e_n\) is the Kronecker delta at \(n\).

One has \(\sum _n \nu (n)/n = 0\)

For any \(x {\gt} 0\), one has \(\psi (x) - \psi (x/6) \leq U(x)\).

For any \(x {\gt} 0\), one has \(\psi (x) \geq U(x)\).

If \(\nu : \mathbb {N}\to \mathbb {R}\) is finitely supported, then

For \(x \geq 0\), we have \(T(x) = \sum _{n \leq x} \Lambda (n) \lfloor x/n \rfloor \).

For \(x \geq 1\), we have \(T(x) \leq x \log x - x + 1 - \log x\).

For \(x \geq 1\), we have \(T(x) \leq x \log x - x + 1 + \log x\).

We define \(U(x) := \sum _m \nu (m) T(x/m)\).

One has

If \(a\ (q)\) is a primitive residue class, then one has

The (second) Chebyshev Psi function is defined as

where \(\Lambda (n)\) is the von Mangoldt function.

We say that \(E_\pi \) satisfies a classical bound with parameters \(A, B, C, R, x_0\) if for all \(x \geq x_0\) we have

We say that it obeys a numerical bound with parameter \(ε(x_0)\) if for all \(x \geq x_0\) we have

We say that \(E_ψ\) satisfies a classical bound with parameters \(A, B, C, R, x_0\) if for all \(x \geq x_0\) we have

We say that it obeys a numerical bound with parameter \(ε(x_0)\) if for all \(x \geq x_0\) we have

We say that \(E_\theta \) satisfies a classical bound with parameters \(A, B, C, R, x_0\) if for all \(x \geq x_0\) we have

We say that it obeys a numerical bound with parameter \(ε(x_0)\) if for all \(x \geq x_0\) we have

We say that one has a classical zero-free region with parameter \(R\) if \(zeta(s)\) has no zeroes in the region \(Re(s) \geq 1 - 1 / R * \log |\Im s|\) for \(\Im (s) {\gt} 3\).

Let \(M{\gt}0\) and let \(x\) be a complex number such that \(\Re x\leq M\). Then, \(|x|\leq |2M - x|\).

Conjugation symmetry of the Riemann zeta function. Let \(s \in \mathbb {C}\). Then

Conjugation symmetry of the Riemann zeta function in the half-plane of convergence. Let \(s \in \mathbb {C}\) with \(\Re (s) {\gt} 1\). Then \(\overline{\zeta (\overline{s})} = \zeta (s)\).

For \(x{\gt}1\) and \(\sigma {\lt}-3/2\), we have

For every \(x {\gt} 0\) we have \(\psi (x) = \sum _{k \geqslant 1} \theta (x^{1/k})\).

For every \(x {\gt} 0\) and \(n\) we have \(\psi (x^{1/n}) = \sum _{k \geqslant 1} \theta (x^{1/nk})\).

For every \(x {\gt} 0\) we have

For every \(x {\gt} 0\) we have

For every \(x {\gt} 0\) we have

For every \(x {\gt} 0\) we have

For every \(x {\gt} 0\) we have

For every \(x {\gt} 0\) we have

For every \(x {\gt} 0\) we have \(\psi (x) - \theta (x) \leqslant \psi (x^{1/2}) + \psi (x^{1/3}) + \psi (x^{1/5})\).

For every \(x {\gt} 0\) we have \(\psi (x) - \theta (x) \geqslant \psi (x^{1/2}) + \psi (x^{1/3}) + \psi (x^{1/7})\).

If \(\psi : \mathbb {R}\to \mathbb {C}\) is \(C^2\) and compactly supported with \(f\) and \(\hat\psi \) non-negative, then there exists a constant \(B\) such that

for all \(x {\gt} 0\).

If \(\psi :\mathbb {R}\to \mathbb {C}\) is \(C^2\) and obeys the bounds

for all \(t \in \mathbb {R}\), then

for all \(u \in \mathbb {R}\), where \(C\) is an absolute constant.

If \(\psi :\mathbb {R}\to \mathbb {C}\) is absolutely integrable, absolutely continuous, and \(\psi '\) is of bounded variation, then

for all \(u \in \mathbb {R}\).

We have

for \(\Re (s) {\gt} 1\), where \(\chi \) runs over homomorphisms from \(G\) to \(\mathbb {C}^\times \) and \(L\) is the Artin \(L\)-function.

For any non-principal character \(\chi \) of \(Gal(K/L)\), \(L(\chi ,s)\) does not vanish for \(\Re (s)=1\).

For any non-principal character \(\chi \) of \(Gal(K/L)\),

\(\zeta _L\) has a simple pole at \(s=1\).

For all \(t\in \mathbb {R}\) with \(|t|\geq 2\) we have that

Let \(\nu \) be a bumpfunction supported in \([1/2,2]\). Then for any \(\epsilon {\gt}0\), we define the delta spike \(\nu _\epsilon \) to be the function from \(\mathbb {R}_{{\gt}0}\) to \(\mathbb {C}\) defined by

For any \(\epsilon {\gt}0\), we have

Let \(\delta _t=E/\log |t|\) where \(E\) is the constant coming from Theorem 4.3.10.

Let \(f : \mathbb {C} \to \mathbb {C}\) be a function at \(p \in \mathbb {C}\) with derivative \(a\). Then the derivative of the function \(g(z) = \overline{f(\overline{z})}\) at \(\overline{p}\) is \(\overline{a}\).

Conjugation symmetry of the derivative of the Riemann zeta function. Let \(s \in \mathbb {C}\). Then

Let \(R,\, M{\gt}0\) and \(0 {\lt} r {\lt} r' {\lt} R\). Let \(f\) be analytic on \(|z|\leq R\) such that \(f(0)=0\) and suppose \(\Re f(z)\leq M\) for all \(|z|\leq R\). Then we have that

for all \(|z|\leq r\).

Let \(R,\, M{\gt}0\) and \(0 {\lt} r {\lt} r' {\lt} R\). Let \(f\) be analytic on \(|z|\leq R\) such that \(f(0)=0\) and suppose \(\Re f(z)\leq M\) for all \(|z|\leq R\). Then we have that

for all \(|z|\leq r\).

For any \(s = \sigma + tI \in \mathbb {C}\), \(1/2 \le \sigma \le 2, 3 {\lt} |t|\), and any \(0 {\lt} A {\lt} 1\) sufficiently small, and \(1-A/\log |t| \le \sigma \), we have

Let \(x{\gt}0\). Then for \(0 {\lt} c {\lt} 1 /2\), we have that the function

is bounded above on the rectangle with corners at \(-1-c-i*c\) and \(-1+c+i*c\) (except at \(s=-1\)).

Let \(x{\gt}0\). Then for \(0 {\lt} c {\lt} 1 /2\), we have that the function

is bounded above on the rectangle with corners at \(-c-i*c\) and \(c+i*c\) (except at \(s=0\)).

The difference of two vertical integrals and a rectangle is the difference of an upper and a lower U integrals.

Any primitive residue class contains an infinite number of primes.

Let \(B{\gt}1\) and \(0 {\lt} R{\lt}1\). If \(f:\mathbb {C}\to \mathbb {C}\) is a function analytic on neighborhoods of points in \(\overline{\mathbb {D}_1}\) with \(|f(z)|\leq B\) for \(|z|\leq R\), then \(|B_f(z)|\leq B\) for \(|z|\leq R\) also.

There exist integers \(m \ge 0\) and \(r\) satisfying \(0 {\lt} r {\lt} 4 p_1 p_2 p_3\) and

Given a complex function \(f\), we define the function

Let \(f\) be a complex function and let \(z\neq 0\). Then, with \(g\) defined as in Definition 4.3.1,

For \(\sigma _0 {\gt} 1\), there exists a constant \(C {\gt} 0\) such that

For \(x \geq 17\), we have

For \(x {\gt} 1\), we have

For \(x \geq 599\), we have

For \(x {\gt} 1\), we have

For \(x \geq 88789\), we have

For \(x {\gt} 1\), we have

For \(x \geq 5393\), we have

For \(x {\gt} e^{1.112}\), we have

For \(x \geq 468049\), we have

For \(x {\gt} 5.6\), we have

For all \(x \geq 468991632\), there exists a prime \(p\) such that

For \(x {\gt} 0\), we have

We have \(\vartheta (x)\) and \(\psi (x)-\vartheta (x)\) for various \(x\) given by [ 12 , Table 2 ] , in particular \(\vartheta (10^{15}) = 999999965752660.939839805291048\dots \)

We have for \(k \geq 4\), \(p_k \leq k \log p_k\).

We have for \(k \geq 2\), \(\log p_k \leq \log k + \log \log k + 1\).

We have for \(k \geq 178974\),

For \(x \geq e^{b}\) we have \(\psi (x) - x| \leq \varepsilon \), where \(b, \varepsilon \) are given by [ 12 , Table 1 ] .

For \(x \geq 121\), we have

For \(x {\gt} 0\), we have

We have for \(p_k \geq 10^{11}\),

We have for \(p_k \geq 10^{15}\),

We have for \(k \geq 781\),

We have for \(k \geq 688383\),

We have for \(k \geq 3\),

There exists a constant \(X_0\) (one may take \(X_0 = 89693\)) with the following property: for every real \(x \geq X_0\), there exists a prime \(p\) with

We have \(|\psi (x) - x| {\lt} \eta _k x / \log ^k x\) for \(x \geq 2\) with \((k,\eta )\) given by the table in [ 12 , Theorem 3.3 ] .

We have \(|\vartheta (x) - x| {\lt} \eta _k x / \log ^k x\) for \(x \geq x_k\) with \((k,\eta _k,x_k)\) given by the table in [ 12 , Theorem 4.2 ] .

For \(x \geq 4 \times 10^9\), we have

We have for \(x \geq 2278383\),

We have for \(x \geq 912560\),

We have for \(x \geq 2278382\),

We have for \(x \geq 2278382\),

We recall that \(e(\alpha ) = e^{2\pi i \alpha }\).

\(E_\pi (x) = |\pi (x) - \mathrm{Li}(x)| / \mathrm{Li}(x)\)

\(E_ψ(x) = |ψ(x) - x| / x\)

One can find a balanced factorization of \(n!\) with cardinality at most \(n - n / \log n + o(n / \log n)\).-

One can factorize \(n!\) into at most \(n/2 - n / 2\log n + o(n / \log n)\) numbers of size at most \(n^2\).-

\(E_\theta (x) = |\theta (x) - x| / x\)

If \(f\) is differentiable on a set \(s\) except at \(c\in s\), and \(f\) is bounded above on \(s\setminus \{ c\} \), then there exists a differentiable function \(g\) on \(s\) such that \(f\) and \(g\) agree on \(s\setminus \{ c\} \).

\(F_{\pm , \lambda }\) is the Fourier transform of \(\varphi _{\pm , \lambda }\).

\(F\) is absolutely integrable.

\(F_{+,\lambda }(y) \geq I_\lambda (y)\) for all \(y\).

\(F_{+,\lambda }(y) \geq I_\lambda (y)\) for all \(y\).

\(\int (I_\lambda (y) - F_{-,\lambda }(y))\ dy = \frac{1}{|\lambda |} - \frac{1}{e^{|\lambda |} - 1}\).

\(\int (F_{+,\lambda }(y)-I_\lambda (y))\ dy = \frac{1}{1-e^{-|\lambda |}} - \frac{1}{|\lambda |}\).

\(F_{\pm ,\lambda }\) is real-valued.

We work with (approximate) factorizations \(a_1 \dots a_t\) of a factorial \(n!\).

Let \(B{\gt}1\) and \(0 {\lt} r' {\lt} r {\lt} R' {\lt} R{\lt}1\). If \(f:\mathbb {C}\to \mathbb {C}\) is a function analytic on neighborhoods of points in \(\overline{\mathbb {D}_1}\) with \(f(0)=1\) and \(|f(z)|\leq B\) for all \(|z|\leq R\), then for all \(z\in \overline{\mathbb {D}_{R'}}\setminus \mathcal{K}_f(R')\) we have

For any arithmetic function \(f\) and real number \(x\), one has

and

If \(\psi : \mathbb {R}\to \mathbb {C}\) is integrable and \(x {\gt} 0\), then for any \(\sigma {\gt}1\)

For each \(\sigma _1, \sigma _2, \tilde c_1, \tilde c_2\) given in Table 8, we have \(N(\sigma ,T) \leq \tilde c_1 T^{p(\sigma )} \log ^{q(\sigma )} + \tilde c_2 \log ^2 T\) for \(\sigma _1 \leq \sigma \leq \sigma _2\) with \(p(\sigma ) = 8/3 (1-\sigma )\) and \(q(σ) = 5-2\sigma \).

If \(\sigma _1 \geq 0.9\) then \(\Sigma _{\sigma _1}^{\sigma _2} \leq 0.00125994 x^{\sigma _2-1}\).

Let \(5/8 {\lt} \sigma _2 \leq 1\), \(t_0 = t_0(\sigma _2, x) = \max \left(H_{\sigma _2}, \exp \left(\sqrt{\frac{\log x}{R}}\right)\right)\), \(T {\gt} t_0\). Let \(K \geq 2\), \(\lambda = (T/t_0)^{1/K}\), and consider \((t_k)_{k=0}^K\) the sequence given by \(t_k = t_0 \lambda ^k\). Then

where

and \(\tilde{N}(\sigma , T)\) satisfy (ZDB) \(N(\sigma , T) \leq \tilde{N}(\sigma , T)\).

For each pair \(T_0,S_0\) in Table 1 we have, for all \(V {\gt} T_0\), \(\sum _{0 {\lt} \gamma {\lt} V} 1/\gamma {\lt} S_0 + B_1(T_0,V)\).

\(\Sigma _a^b = 2 * \sum _{H_a ≤ \gamma {\lt} T; a \leq \beta {\lt} b} \frac{x^{\beta -1}}{\gamma }\).

If \(|N(T) - (T/2\pi \log (T/2\pi e) + 7/8)| \leq R(T)\) then \(\sum _{U \leq \gamma {\lt} V} 1/\gamma \leq B_1(U,V)\).

Let \(T_0 \geq 2\) and \(\gamma {\gt} 0\). Assume that there exist \(c_1, c_2, p, q, T_0\) for which one has a zero density bound. Assume \(\sigma \geq 5/8\) and \(T_0 \leq U {\lt} V\). Then \(s_0(σ,U,V) \leq B_0(\sigma ,U,V)\).

For all \(0 {\lt} \log x \leq 2100\) we have that

For all \(2100 {\lt} \log x \leq 200000\) we have that

Let \(5/8 {\lt} \sigma _2 \leq 1\), \(t_0 = t_0(\sigma _2,x) = \max (H_{\sigma _2}, \exp ( \sqrt{\log x}/R))\) and \(T {\gt} 0\). Let \(K \geq 2\) and consider a strictly increasing sequence \((t_k)_{k=0}^K\) such that \(t_k = T\). Then \(\Sigma _{\sigma _2}^1 ≤ 2 N(\sigma _2,T) x^{-1/R\log t_0}/t_0\) and \(\Sigma _{\sigma _2}^1 ≤ 2 ((\sum _{k=1}^{K-1} N(\sigma _2, t_k) (x^{-1/R\log t_{k-1}} / t_{k-1} - x^{-1/(R \log t_k)}/t_k)) + x^{-1/R \log t_{K-1}}/t_{K-1} N(\sigma _2,T))\).

Fix \(K \geq 2\) and \(c {\gt} 1\), and set \(t_0\), \(T\), and \(\sigma _2\) as functions of \(x\) defined by

Then, with \(\varepsilon _4(x, \sigma _2, K, T)\) as defined in (3.22), we have that as \(x \to \infty \),

where \(c_1\) is an admissible value for (ZDB) on some interval \([\sigma _1, 1]\). Moreover, both \(\varepsilon _4(x, \sigma _2, K, T)\) and \(\frac{\varepsilon _4(x, \sigma _2, K, T) t_0^2}{(\log t_0)^3}\) are decreasing in \(x\) for \(x {\gt} \exp (Re^2)\).

Let \(x {\gt} e^{50}\) and \(3 \log x {\lt} T {\lt} \sqrt{x}/3\). Then \(E_\psi (x) ≤ \sum _{|\gamma | {\lt} T} |x^{\rho -1}/\rho | + 2 \log ^2 x / T\).

Let \(\sigma _1 \in (1/2,1)\) and let \((T_0,S_0)\) be taken from Table 1. Then \(\Sigma _0^{\sigma _1} ≤ 2 x^{-1/2} (S_0 + B_1(T_0,T)) + (x_1^{\sigma _1-1} - x^{-1/2}) B_1(H_0,T)\).

Let \(N \geq 2\) be an integer. If \(5/8 \leq \sigma _1 {\lt} \sigma _2 \leq 1\), \(T \geq H_0\), then \(\Sigma _{\sigma _1}^{\sigma _2} ≤ 2 x^{-(1-\sigma _1)+(\sigma _2-\sigma _1/N)}B_0(\sigma _1, H_{\sigma _1}, T) + 2 x^{(1-\sigma _1)} (1 - x^{-(\sigma _2-\sigma _1)/N}) \sum _{n=1}^{N-1} B_0(\sigma ^{(n)}, H^{(n)}, T) x^{(\sigma _2-\sigma _1) (n+1)/N}\).

\(\Gamma (3,x) = (x^2 + 2(x+1)) e^{-x}\).

For \(s{\gt}1\), one has \(\Gamma (s,x) \sim x^{s-1} e^{-x}\).

If \(\sigma {\lt} 1 - 1/R \log H_0\) then \(H_σ = H_0\).

For any \(x_0\) with \(\log x_0 {\gt} 1000\), and all \(0.9 {\lt} \sigma _2 {\lt} 1\), \(2 \leq c \leq 30\), and \(N, K \geq 1\) the formula \(\varepsilon (x_0) := \varepsilon (x_0, \sigma _2, c, N, K)\) as defined in (4.1) gives an effectively computable bound

Moreover, a collection of values, \(\varepsilon (x_0)\) computed with well chosen parameters are provided in Table 5.

If \(\log x_0 \geq 1000\) then we have an admissible bound for \(E_\psi \) with the indicated choice of \(A(x_0)\), \(B = 3/2\), \(C = 2\), and \(R = 5.5666305\).

Let \(H_0\) denote a verification height for RH. Let \(10^9/H_0≤ k \leq 1\), \(t {\gt} 0\), \(H \in [1002, H_0)\), \(α {\gt} 0\), \(δ ≥ 1\), \(\eta _0 = 0.23622\), \(1 + \eta _0 \leq \mu \leq 1+\eta \), and \(\eta \in (\eta _0, 1/2)\) be fixed. Let \(\sigma {\gt} 1/2 + d / \log H_0\). Then for any \(T \geq H_0\), one has

and

.

Let \(x {\gt} e^{50}\) and \(50 {\lt} T {\lt} x\). Then \(E_\psi (x) \leq \sum _{|\gamma | {\lt} T} |x^{\rho -1}/\rho | + 2 \log ^2 x / T\).

For any \(\alpha \in (0,1/2]\) and \(\omega \in [0,1]\) there exist \(M, x_M\) such that for \(\max (51, \log x) {\lt} T {\lt} (x^\alpha -2)/5\) and some \(T^* \in [T, 2.45 T]\),

for all \(x ≥ x_M\).

If \(B \geq 1 + C^2 / 16R\) then \(g(1,1-B,C/\sqrt{R},x)\) is decreasing in \(x\).

We have an admissible bound for \(E_\theta \) with \(A = 121.0961\), \(B=3/2\), \(C=2\), \(R = 5.5666305\), \(x_0=2\).

Let \(B \geq \max (\frac{3}{2}, 1 + \frac{C^2}{16R})\) and \(B {\gt} C^2/8R\). Let \(x_0, x_1 {\gt} 0\) with \(x_1 \geq \max (x_0, \exp ( (1 + \frac{C}{2\sqrt{R}})^2))\). If \(E_\psi \) satisfies an admissible classical bound with parameters \(A_\psi ,B,C,R,x_0\), then \(E_\pi \) satisfies an admissible classical bound with \(A_\pi , B, C, R, x_1\), where

for all \(x \geq x_0\), where

where

and

One has

for all \(x \geq 2\).

\(A_\pi , B, C, x_0\) as in [ 14 , Table 6 ] give an admissible asymptotic bound for \(E_\pi \) with \(R = 5.5666305\).

We have the bounds \(E_\pi (x) \leq B(x)\), where \(B(x)\) is given by Table 7.

One has

for all \(x \geq 2\).

Let \(\{ b'_i\} _{i=1}^M\) be a set of finite subdivisions of \([\log (x_1),\infty )\), with \(b'_1 = \log (x_1)\) and \(b'_M = \infty \). Define

Then \(E_\pi (x) \leq \varepsilon _{\pi ,num}(x_1)\) for all \(x \geq x_1\).

For \(x_1 \leq x_2 \leq x_1 \log x_1\), we define

For \(x_2 \geq x_1 \log x_1\), we define

For \(x_1 \leq x_2\), we define \(\epsilon _{\pi ,num}(x_0,x_1,x_2) := \epsilon _{\theta ,num}(x_1) (1 + \mu _{num}(x_0,x_1,x_2))\).

For any \(a,b,c,x \in \mathbb {R}\) we define \(g(a,b,c,x) := x^{-a} (\log x)^b \exp ( c (\log x)^{1/2} )\).

For any \(2 \leq x_0 {\lt} x\) one has

The Dawson function \(D_+ : \mathbb {R} \to \mathbb {R}\) is defined by the formula \(D_+(x) := e^{-x^2} \int _0^x e^{t^2}\ dt\).

For \(x_0,x_1 {\gt} 0\), we define

.

For any \(x \geq x_0 {\gt} 0\),

For any \(x_0{\gt}0\), we define

We have

\(\frac{d}{dx} g(a, b, c, x) \) is negative when \(-au^2 + \frac{c}{2}u + b {\lt} 0\), where \(u = \sqrt{\log (x)}\).

If \(a{\gt}0\), \(c{\gt}0\) and \(b {\lt} -c^2/16a\), then \(g(a,b,c,x)\) decreases with \(x\).

For any \(a{\gt}0\), \(c{\gt}0\) and \(b \geq -c^2/16a\), \(g(a,b,c,x)\) decreases with \(x\) for \(x {\gt} \exp ((\frac{c}{4a} + \frac{1}{2a} \sqrt{\frac{c^2}{4} + 4ab})^2)\).

If \(c{\gt}0\), \(g(0,b,c,x)\) decreases with \(x\) for \(\sqrt{\log x} {\lt} -2b/c\).

Suppose that \(E_\theta \) satisfies an admissible classical bound with parameters \(A,B,C,R,x_0\). Then, for all \(x \geq x_0\),

where

Let \(x_1 {\gt} x_0 \geq 2\), \(N \in \mathbb {N}\), and let \((b_i)_{i=1}^N\) be a finite partition of \([x_0,x_1]\). Then

The function \(\operatorname {Li}(x) - \frac{x}{\log x}\) is strictly increasing for \(x {\gt} 6.58\).

Assume \(x {\gt} 6.58\). Then \(\operatorname {Li}(x) - \frac{x}{\log x} {\gt} \frac{x-6.58}{\log ^2 x} {\gt} 0\).

We define \(\mu _{num}(x_0, x_1, x_2)\) to equal \(\mu _{num,1}(x_0,x_1,x_2)\) when \(x_2 \leq x_1 \log x_1\) and \(\mu _{num,2}(x_0,x_1)\) otherwise.

Suppose that \(A_\psi ,B,C,R,x_0\) give an admissible bound for \(E_\psi \). If \(B {\gt} C^2/8R\), then \(A_\theta , B, C, R, x_0\) give an admissible bound for \(E_\theta \), where

with

and \(a_1,a_2\) are given by Definitions 9.7.4 and 9.7.5.

Let \(x {\gt} x_0 {\gt} 2\). If \(E_\psi (x) \leq \varepsilon _{\psi ,num}(x_0)\), then

where

If \(B \geq \max (3/2, 1 + C^2/16 R)\), \(x_0 {\gt} 0\), and one has an admissible asymptotic bound with parameters \(A,B,C,x_0\) for \(E_\theta \), and

then

for all \(x \geq x_1\). In other words, we have an admissible bound with parameters \((1+\mu _{asymp}(x_0,x_1))A, B, C, x_1\) for \(E_\pi \).

Let \(x_0 {\gt} 0\) be chosen such that \(\pi (x_0)\) and \(\theta (x_0)\) are computable, and let \(x_1 \geq \max (x_0, 14)\). Let \(\{ b_i\} _{i=1}^N\) be a finite partition of \([\log x_0, \log x_1]\), with \(b_1 = \log x_0\) and \(b_N = \log x_1\), and suppose that \(\varepsilon _{\theta ,\mathrm{num}}\) gives computable admissible numerical bounds for \(x = \exp (b_i)\), for each \(i=1,\dots ,N\). For \(x_1 \leq x_2 \leq x_1 \log x_1\), we define

and for \(x_2 {\gt} x_1 \log x_1\), including the case \(x_2 = \infty \), we define

Then, for all \(x_1 \leq x \leq x_2\) we have

With the above hypotheses, for all \(x \geq x_1\) we have

With the above hypotheses, for all \(x \geq x_1\) we have

With the above hypotheses, for all \(x \geq x_1\) we have

For all x > 2 we have \(E_ψ(x) \leq 121.096 (\log x/R)^{3/2} \exp (-2 \sqrt{\log x/R})\) with \(R = 5.5666305\).

For all x > 2 we have \(E_ψ(x) \leq 9.22022(\log x)^{3/2} \exp (-0.8476836 \sqrt{\log x})\).

For \(x{\gt}1\) and \(\sigma {\gt}0\), we have

For \(x{\gt}0\), \(\sigma {\gt}0\), and \(x{\lt}1\), we have

\(\theta (x) {\lt} x\) for all \(1 \leq x \leq 10^{19}\).

For \(\sigma {\gt}1\) and \(t\in \mathbb {R}\), define

Then \(F\) is real-valued, and whence \(F(\sigma )\ge 1\) there.

Let \(t\in \mathbb {R}\) with \(|t|\geq 3\) and \(z=\sigma +it\) where \(1-\delta _t/3\leq \sigma \leq 3/2\). Additionally, let \(\rho \in \mathcal{Z}_t\). Then we have that

For all \(s\in \mathbb {C}\) with \(|s|\leq 1\) and \(t\in \mathbb {R}\) with \(|t|\geq 2\), we have that

Let \(f : \mathbb {C} \to \mathbb {C}\) be a complex differentiable function at \(p \in \mathbb {C}\) with derivative \(a\). Then the function \(g(z) = \overline{f(\overline{z})}\) is complex differentiable at \(\overline{p}\) with derivative \(\overline{a}\).

A positive integer \(N\) is called highly abundant (HA) if

for all positive integers \(m {\lt} N\), where \(\sigma (n)\) denotes the sum of the positive divisors of \(n\).

If \(f\) is holomorphic on a rectangle \(z\) and \(w\), then the integral of \(f\) over the rectangle with corners \(z\) and \(w\) is \(0\).

For any \(N\ge 1\), the function \(\zeta _0(N,s)\) is holomorphic on \(\{ s\in \mathbb {C}\mid \Re (s){\gt}0 ∧ s \ne 1\} \).

We have that

Same with \(I_9\).

Let

We have that

Assuming a bound of the form of Lemma 3.4.28 we have that

Let

We have that

Assuming a bound of the form of Lemma 3.4.28 we have that

Same with \(I_7\).

Let

We have that

We have that

Same with \(I_6\).

Let

We have that

We have that

Let

We have that

We have that

Symmetry between \(I_2\) and \(I_8\):

Given a natural number \(k\) and a real number \(X_0 {\gt} 0\), there exists \(C \geq 1\) so that for all \(X \geq X_0\),

The number of elements in this initial factorization is at most \(n\).

We perform an initial factorization by taking the natural numbers between \(n-n/M\) (inclusive) and \(n\) (exclusive) repeated \(M\) times, deleting those elements that are not \(n/L\)-smooth (i.e., have a prime factor greater than or equal to \(n/L\)).

A large prime \(p \geq n/L\) can be in deficit by at most \(n/p\).

A large prime \(p \geq n/L\) cannot be in surplus.

A medium prime \(\sqrt{n} {\lt} p ≤ n/L\) can be in deficit by at most \(M\).

A medium prime \(\sqrt{n} {\lt} p ≤ n/L\) can be in surplus by at most \(M\).

A small prime \(L {\lt} p \leq \sqrt{n}\) can be in deficit by at most \(M\log n\).

A small prime \(p \leq \sqrt{n}\) can be in surplus by at most \(M\log n\).

A tiny prime \(p \leq L\) can be in deficit by at most \(M\log n + ML\pi (n)\).

The total waste in this initial factorization is at most \(n \log \frac{1}{1-1/M}\).

The score of the initial factorization can be taken to be \(o(n)\).

The initial score is bounded by

The integral

is positive (and hence convergent - since a divergent integral is zero in Lean, by definition).

The conjugation symmetry of the interval integral. Let \(f : \mathbb {R} \to \mathbb {C}\) be a measurable function, and let \(a, b \in \mathbb {R}\). Then

Additive function.

Completely additive function.

Let \(x{\gt}0\). Then the function \(f(s) = x^s/(s(s+1))\) is holomorphic on the half-plane \(\{ s\in \mathbb {C}:\Re (s){\gt}0\} \).

Let \(x{\gt}0\) and \(\sigma \in \mathbb {R}\). Then

is integrable.

The set \(\{ s\in \mathbb {C}\mid \Re (s){\gt}0 ∧ s \ne 1\} \) is path-connected.

Let \(B{\gt}1\) and \(0 {\lt} R{\lt}1\). If \(f:\mathbb {C}\to \mathbb {C}\) is a function analytic on neighborhoods of points in \(\overline{\mathbb {D}_1}\) with \(f(0)=1\), define \(L_f(z)=J_{B_f}(z)\) where \(J\) is from Theorem 4.3.3 and \(B_f\) is from Definition 4.3.7.

Let \(B{\gt}1\) and \(0 {\lt} r' {\lt} r {\lt} R{\lt}1\). If \(f:\mathbb {C}\to \mathbb {C}\) is a function analytic on neighborhoods of points in \(\overline{\mathbb {D}_1}\) with \(f(0)=1\) and \(|f(z)|\leq B\) for all \(|z|\leq R\), then for all \(|z|\leq r'\)

Let \(x\in \mathbb {R}\) and \(s \ne 0, -1\). Then

We have \(\sum _{n \leq x} \lambda (n) = o(x)\).

In the next few subsections we assume that \(n \geq 1\) and that \(p_1,p_2,p_3,q_1,q_2,q_3\) are primes satisfiying

and the key criterion

NOTE: In the Lean formalization of this argument, we index the primes from 0 to 2 rather than from 1 to 3.

With \(m,r\) as above, define the competitor

We have \(4 p_1 p_2 p_3 {\lt} q_1 q_2 q_3\).

Let \(s = \sigma + i \tau \), \(\sigma ,\tau \in \mathbb {R}\). Let \(n\in \mathbb {Z}_{{\gt}0}\). Let \(a,b\in \mathbb {Z} + \frac{1}{2}\), \(b{\gt}a{\gt}\frac{|\tau |}{2\pi n}\). Write \(\varphi _\nu (t) = \nu t - \frac{\tau }{2\pi } \log t\). Then

Let \(\sigma \geq 0\), \(\tau \in \mathbb {R}\), \(\nu \in \mathbb {R}\setminus \{ 0\} \). Let \(b{\gt}a{\gt}\frac{|\tau |}{2\pi |\nu |}\). Then, for any \(k\geq 1\), \(f(t) = t^{-\sigma -k} |2\pi \nu -\tau /t|^{-k-1}\) is decreasing on \([a,b]\).

Let \(s = \sigma + i \tau \), \(\sigma \geq 0\), \(\tau \in \mathbb {R}\). Let \(\nu \in \mathbb {R}\setminus \{ 0\} \), \(b{\gt}a{\gt}\frac{|\tau |}{2\pi |\nu |}\). Then

where \(\varphi _\nu (t) = \nu t - \frac{\tau }{2\pi } \log t\) and \(\vartheta = \frac{\tau }{2\pi a}\).

Let \(s = \sigma + i \tau \), \(\sigma \geq 0\), \(\tau \in \mathbb {R}\). Let \(\nu \in \mathbb {R}\setminus \{ 0\} \), \(b{\gt}a{\gt}\frac{|\tau |}{2\pi |\nu |}\). Then

where \(\varphi _\nu (t) = \nu t - \frac{\tau }{2\pi } \log t\).

Let \(\varphi :[a,b]\to \mathbb {R}\) be \(C^1\) with \(\varphi '(t)\ne 0\) for all \(t\in [a,b]\). Let \(h:[a,b]\to \mathbb {R}\) be such that \(g(t) = h(t)/\varphi '(t)\) is continuous and \(|g(t)|\) is non-increasing. Then

Let \(g:[a,b]\to \mathbb {R}\) be continuous, with \(|g(t)|\) non-increasing. Then \(g\) is monotone, and \(\| g\| _{\mathrm{TV}} = |g(a)|-|g(b)|\).

Let \(z\in \mathbb {C}\), \(z\notin \mathbb {Z}\). Then

Let \(b{\gt}0\), \(b\in \mathbb {Z} + \frac{1}{2}\). Then, for all \(s\in \mathbb {C}\setminus \{ 1\} \) with \(\Re s {\gt} 0\),

Let \(b{\gt}0\), \(b\in \mathbb {Z}\). Then, for all \(s\in \mathbb {C}\setminus \{ 1\} \) with \(\Re s {\gt} 0\),

Let \(z\in \mathbb {C}\), \(z\notin \mathbb {Z}\). Then

For \(\vartheta \in \mathbb {R}\) with \(0\leq |\vartheta |{\lt} 1\),

Let \(s = \sigma + i \tau \), \(\sigma \geq 0\), \(\tau \in \mathbb {R}\), with \(s\ne 1\). Let \(b{\gt}a{\gt}0\), \(a, b\in \mathbb {Z} + \frac{1}{2}\), with \(a{\gt}\frac{|\tau |}{2\pi }\). Define \(f:\mathbb {R}\to \mathbb {C}\) by \(f(y) = 1_{[a,b]}(y)/y^s\). Write \(\vartheta = \frac{\tau }{2\pi a}\), \(\vartheta _- = \frac{\tau }{2\pi b}\). Then

with absolute convergence, where \(g(t) = \frac{1}{\sin \pi t} - \frac{1}{\pi t}\) for \(t\ne 0\), \(g(0)=0\), and

Let \(b{\gt}a{\gt}0\), \(b\in \mathbb {Z} + \frac{1}{2}\). Then, for all \(s\in \mathbb {C}\setminus \{ 1\} \) with \(\sigma = \Re s {\gt} 0\),

Let \(a,b\in \mathbb {R}\setminus \mathbb {Z}\), \(b{\gt}a{\gt}0\). Let \(s\in \mathbb {C}\setminus \{ 1\} \). Define \(f:\mathbb {R}\to \mathbb {C}\) by \(f(y) = 1_{[a,b]}(y)/y^s\). Then

Let \(n \ge X_0^2\). Set \(x := \sqrt{n}\). Then there exist primes \(p_1,p_2,p_3\) with

and \(p_1 {\lt} p_2 {\lt} p_3\). Moreover, \(\sqrt{n} {\lt} p_1\)

Let \(n \ge X_0^2\). Then there exist primes \(q_1 {\lt} q_2 {\lt} q_3\) with

for \(i = 1,2,3\), and \(q_1 {\lt} q_2 {\lt} q_3 {\lt} n\).

If

then \(L_n\) is not highly abundant.

Suppose

Then \(\sigma (M) \ge \sigma (L_n)\), and so \(L_n\) is not highly abundant.

For all \(n \ge X_0^2 = 89693^2\) we have

and

For \(0 \le \varepsilon \le 1/89693^2\), we have

There exists a positive integer \(L'\) such that

and each prime \(q_i\) divides \(L_n\) exactly once and does not divide \(L'\).

With notation as above, we have:

1. \(M {\lt} L_n\).

2. \[ 1 {\lt} \frac{L_n}{M} = \Bigl(1 - \frac{r}{q_1 q_2 q_3}\Bigr)^{-1} {\lt} \Bigl(1 - \frac{4 p_1 p_2 p_3}{q_1 q_2 q_3}\Bigr)^{-1}. \]

With \(p_i\) as in Lemma 10.1.9, we have for large \(n\)

For \(0 \le \varepsilon \le 1/89693^2\), we have

and

With \(p_i,q_i\) as in Lemmas 10.1.9 and 10.1.10, we have

With \(p_i,q_i\) as in Lemmas 10.1.9 and 10.1.10, we have

Let \(L'\) be as in Lemma 10.1.2. Then

With notation as above,

\(\mathrm{li}(x) = \int _0^x \frac{dt}{\log t}\) (in the principal value sense) and \(\mathrm{Li}(x) = \int _2^x \frac{dt}{\log t}\).

\(1.039 \leq \operatorname {li}(2) \leq 1.06\).

\(\int _0^1 \left(\frac{1}{\log (1+t)} + \frac{1}{\log (1-t)}\right) dt = \mathrm{li}(2)\)

If \(\psi : \mathbb {R}\to \mathbb {C}\) is \(C^2\) and compactly supported and \(x \geq 1\), then

With the hypotheses as above, we have

as \(x \to \infty \).

If \(\psi : \mathbb {R}\to \mathbb {C}\) is \(C^2\) and compactly supported with \(f\) and \(\hat\psi \) non-negative, and \(0 {\lt} x\), then

Let \(a:\mathbb {R}\to \mathbb {C}\) be a function, and let \(\sigma {\gt}0\) be a real number. Suppose that, for all \(\sigma , \sigma '{\gt}0\), we have \(a(\sigma ')=a(\sigma )\), and that \(\lim _{\sigma \to \infty }a(\sigma )=0\). Then \(a(\sigma )=0\).

Let \(a:\mathbb {R}\to \mathbb {C}\) be a function, and let \(\sigma {\lt}-3/2\) be a real number. Suppose that, for all \(\sigma , \sigma '{\gt}0\), we have \(a(\sigma ')=a(\sigma )\), and that \(\lim _{\sigma \to -\infty }a(\sigma )=0\). Then \(a(\sigma )=0\).

For each integer \(n \ge 1\), define

We call \((L_n)_{n \ge 1}\) the least common multiple sequence.

For \(t \geq 0\), one has \(t - \frac{t^2}{2} \leq \log (1+t)\).

For \(0 \leq t \leq t_0 {\lt} 1\), one has \(\frac{t}{t_0} \log (1-t_0) \leq \log (1-t)\).

For every \(n\) there is some absolute constant \(C{\gt}0\) such that

For \(t {\gt} -1\), one has \(\log (1+t) \leq t\).

We define Logan’s function

We define

We have that, for \(\Re (s){\gt}1\),

If \(f\) is holomorphic in a neighborhood of \(p\), and there is a simple pole at \(p\), then \(f'/ f\) has a simple pole at \(p\) with residue \(-1\):

There is an \(A{\gt}0\) so that for \(1-A/\log ^9 |t| \le \sigma {\lt} 1+A/\log ^9 |t|\) and \(3 {\lt} |t|\),

There exist \(A, C {\gt} 0\) such that

whenever \(|t|{\gt}3\) and \(\sigma {\gt} 1 - A/\log |t|^9\).

Let \(t\in \mathbb {R}\) with \(|t|\geq 2\) and \(0 {\lt} r' {\lt} r {\lt} R' {\lt} R{\lt}1\). If \(f(z)=\zeta (z+3/2+it)\), then for all \(z\in \overline{\mathbb {D}_R'}\setminus \mathcal{K}_f(R')\) we have that

There is an \(A{\gt}0\) so that for all \(T{\gt}3\), the function \( \frac{\zeta '}{\zeta }(s) \) is holomorphic on \(\{ 1-A/\log ^9 T \le \Re s \le 2, |\Im s|\le T \} \setminus \{ 1\} \).

There is a \(\sigma _2 {\lt} 1\) so that the function

is holomorphic on \(\{ \sigma _2 \le \Re s \le 2, |\Im s| \le 3 \} \setminus \{ 1\} \).

For \(T{\gt}0\) and \(\sigma '=1-\delta _T/3=1-F/\log T\), if \(|t|\leq T\) then we have that

There exists a constant \(F\in (0,1/2)\) such that for all \(t\in \mathbb {R}\) with \(|t|\geq 3\) one has

where the implied constant is uniform in \(\sigma \).

There exists a constant \(F\in (0,1/2)\) such that for all \(t\in \mathbb {R}\) with \(|t|\geq 3\) one has

where the implied constant is uniform in \(\sigma \).

Let \(0 {\lt} r {\lt} R{\lt}1\). Let \(B:\overline{\mathbb {D}_R}\to \mathbb {C}\) be analytic on neighborhoods of points in \(\overline{\mathbb {D}_R}\) with \(B(z)\neq 0\) for all \(z\in \overline{\mathbb {D}_R}\). Then there exists \(J_B:\overline{\mathbb {D}_r}\to \mathbb {C}\) that is analytic on neighborhoods of points in \(\overline{\mathbb {D}_r}\) such that

• \(J_B(0)=0\)

• \(J_B'(z)=B'(z)/B(z)\)

• \(\log |B(z)|-\log |B(0)|=\Re J_B(z)\)

for all \(z\in \overline{\mathbb {D}_r}\).

A LowerUIntegral of a function \(f\) comes from \(\sigma -i\infty \) up to \(\sigma -iT\), over to \(\sigma '-iT\), and back down to \(\sigma '-i\infty \).

The L-series of \(\tau \) equals the square of the Riemann zeta function for \(\Re (s) {\gt} 1\).

\(M(x)\) is the summatory function of the Möbius function.

We have for \(k \geq 198\),

We have

\(B := \lim _{x \to \infty } \left( \sum _{p \leq x} \frac{1}{p} - \log \log x \right)\).

Let \(f\) and \(g\) be functions from \(\mathbb {R}_{{\gt}0}\) to \(\mathbb {C}\). Then we define the Mellin convolution of \(f\) and \(g\) to be the function \(f\ast g\) from \(\mathbb {R}_{{\gt}0}\) to \(\mathbb {C}\) defined by

Let \(f\) and \(g\) be functions from \(\mathbb {R}_{{\gt}0}\) to \(\mathbb {R}\) or \(\mathbb {C}\), for \(x\neq 0\),

Let \(f\) and \(g\) be functions from \(\mathbb {R}_{{\gt}0}\) to \(\mathbb {C}\) such that

is absolutely integrable on \([0,\infty )^2\). Then

The Mellin transform of \(1_{(0,1]}\) is

For any \(\epsilon {\gt}0\), the Mellin transform of \(\nu _\epsilon \) is

For any \(\epsilon {\gt}0\), we have

As \(\epsilon \to 0\), we have

The Mellin transform of \(\nu \) is

as \(|s|\to \infty \) with \(\sigma _1 \le \Re (s) \le 2\).

Fix \(\epsilon {\gt}0\). Then the Mellin transform of \(\widetilde{1_{\epsilon }}\) is

Given \(0{\lt}\sigma _1\le \sigma _2\), for any \(s\) such that \(\sigma _1\le \mathcal Re(s)\le \sigma _2\), we have

At \(s=1\), we have

\(E := \lim _{x \to \infty } \left( \sum _{p \leq x} \frac{\log p}{p} - \log x \right)\).

For any \(x{\gt}0\),

For any \(x{\gt}0\),

.

For any \(x{\gt}0\) and any integer \(K\geq 0\),

For any \(K \leq x\),

For any \(K \leq x\), for \(f(u) = M(\sqrt{x/u})\),

We have \(\sum _{n \leq x} \mu (n) = o(x)\).

We have \(\sum _{n \leq x} \mu (n)/n = o(1)\).

We have \(|\sum _{n \leq x} \frac{\mu (n)}{n}| \leq 1\).

If a function \(f\) has a simple pole at a point \(p\) with residue \(A \neq 0\), then \(f\) is nonzero in a punctured neighborhood of \(p\).

Now let \(M,L\) be additional parameters with \(n {\gt} L^2\); we also need the minor variant \(\lfloor n/L \rfloor {\gt} \sqrt{n}\).

Let \(f, g\) be once differentiable functions from \(\mathbb {R}_{{\gt}0}\) to \(\mathbb {C}\) so that \(fg'\) and \(f'g\) are both integrable, and \(f\cdot g (x)\to 0\) as \(x\to 0^+,\infty \). Then

Let \(0 {\lt} r {\lt} R{\lt}1\). Let \(f:\overline{\mathbb {D}_R}\to \mathbb {C}\) be analytic on neighborhoods of points in \(\overline{\mathbb {D}_R}\). Define the functon \(I_f:\overline{\mathbb {D}_r}\to \mathbb {C}\) by

Let \(x{\gt}0\) and \(\sigma ',\sigma ''\in \mathbb {R}\). Then

goes to \(0\) as \(t\to \infty \).

where

\(\varphi \) is absolutely continuous.

\(\varphi '\) is of bounded variation.

\(\varphi \) is absolutely integrable.

where

\(\pi (x)\) is the number of primes less than or equal to \(x\).

One has

as \(x \to \infty \).

There exists a function \(c(x)\) such that \(c(x) = o(1)\) as \(x \to \infty \) and

for all \(x\) large enough.

One has

as \(n \to \infty \).

We have \(p_{n+1} - p_n = o(p_n)\) as \(n \to \infty \).

If \(\psi :\mathbb {R}\to \mathbb {C}\) is absolutely integrable then

for all \(u \in \mathbb {R}\). where \(C\) is an absolute constant.

If \(\psi :\mathbb {R}\to \mathbb {C}\) is absolutely integrable and of bounded variation, then

for all non-zero \(u \in \mathbb {R}\).

If \(\psi :\mathbb {R}\to \mathbb {C}\) is absolutely integrable, absolutely continuous, and \(\psi '\) is of bounded variation, then

for all non-zero \(u \in \mathbb {R}\).

For every \(\varepsilon {\gt}0\), there is a prime between \(x\) and \((1+\varepsilon )x\) for all sufficiently large \(x\).

For all \(n \geq 2\), one has

We have

Let \(s = \sigma + i \tau \), \(\sigma \geq 0\), \(\tau \in \mathbb {R}\). Let \(a,b\in \mathbb {Z} + \frac{1}{2}\), \(b{\gt}a{\gt}\frac{|\tau |}{2\pi }\). Write \(\vartheta = \frac{\tau }{2\pi a}\). Then, for any integer \(n\geq 1\),

Let \(s = \sigma + i \tau \), \(\sigma \geq 0\), \(\tau \in \mathbb {R}\), with \(s\ne 1\). Let \(a\in \mathbb {Z} + \frac{1}{2}\) with \(a{\gt}\frac{|\tau |}{2\pi }\). Then

where \(\vartheta = \frac{\tau }{2\pi a}\), \(c_\vartheta = \frac{i}{2} \left(\frac{1}{\sin \pi \vartheta } - \frac{1}{\pi \vartheta }\right)\) for \(\vartheta \ne 0\), \(c_0 =0\), and \(C_{\sigma ,\vartheta }\) is as in 4.

For every integer \(n \ge X_0^2 = 89693^2 \approx 8.04\times 10^9\), the primes \(p_i,q_i\) constructed in Lemmas 10.1.9 and 10.1.10 satisfy the inequality 1.

For \(x \geq 30\), we have \(\psi (x) - \psi (x/6) \leq ax + 5\log x - 5\).

For \(x \geq 30\), we have \(\psi (x) \geq ax - 5\log x - 1\).

For \(0 {\lt} x \leq 30\), we have \(\psi (x) \leq 1.015 x\).

For \(0 {\lt} x \leq 11723\), we have \(\psi (x) \leq 1.11 x\).

For \(x \geq 30\), we have \(\psi (x) \leq 6ax/5 + (\log (x/5) / \log 6) (5 \log x - 5)\).

For \(x {\gt} 0\), we have \(\psi (x) \leq 1.11 x\).

\(Q(x)\) is the number of squarefree integers \(\leq x\).

\(R(x) = Q(x) - x / \zeta (2)\).

\(\operatorname {li}(2) = 1.0451\dots \).

A Rectangle’s border, given corners \(z\) and \(w\) is the union of the four sides.

A RectangleIntegral of a function \(f\) is one over a rectangle determined by \(z\) and \(w\) in \(\mathbb {C}\). We will sometimes denote it by \(\int _{z}^{w} f\). (There is also a primed version, which is \(1/(2\pi i)\) times the original.)

Let \(\sigma ,\sigma ' \in \mathbb {R}\), and \(f : \mathbb {C} \to \mathbb {C}\) such that the vertical integrals \(\int _{(\sigma )}f(s)ds\) and \(\int _{(\sigma ')}f(s)ds\) exist and the horizontal integral \(\int _{(\sigma )}^{\sigma '}f(x + yi)dx\) vanishes as \(y \to -\infty \). Then the limit of rectangle integrals

as \(U\to \infty \) is the “LowerUIntegral” of \(f\).

Let \(\sigma ,\sigma ' \in \mathbb {R}\), and \(f : \mathbb {C} \to \mathbb {C}\) such that the vertical integrals \(\int _{(\sigma )}f(s)ds\) and \(\int _{(\sigma ')}f(s)ds\) exist and the horizontal integral \(\int _{(\sigma )}^{\sigma '}f(x + yi)dx\) vanishes as \(y \to \pm \infty \). Then the limit of rectangle integrals

as \(U\to \infty \) is the “UpperUIntegral” of \(f\).

Let \(\sigma ,\sigma ' \in \mathbb {R}\), and \(f : \mathbb {C} \to \mathbb {C}\) such that the vertical integrals \(\int _{(\sigma )}f(s)ds\) and \(\int _{(\sigma ')}f(s)ds\) exist and the horizontal integral \(\int _{(\sigma )}^{\sigma '}f(x + yi)dx\) vanishes as \(y \to \pm \infty \). Then the limit of rectangle integrals

If \(f\) is holomorphic on a rectangle \(z\) and \(w\) except at a point \(p\), then the integral of \(f\) over the rectangle with corners \(z\) and \(w\) is the same as the integral of \(f\) over a small square centered at \(p\).

Let \(x{\gt}0\). Then for all sufficiently small \(c{\gt}0\), we have that

Let \(x{\gt}0\). Then for all sufficiently small \(c{\gt}0\), we have that

If \(f\) has a simple pole at \(p\) with residue \(A\), and \(g\) is holomorphic near \(p\), then the residue of \(f \cdot g\) at \(p\) is \(A \cdot g(p)\). That is, we assume that

near \(p\), and that \(g\) is holomorphic near \(p\). Then

If a function \(f\) is holomorphic in a neighborhood of \(p\) and \(\lim _{s\to p} (s-p)f(s) = A\), then \(f(s) = \frac{A}{s-p} + O(1)\) near \(p\).

For \(x{\gt}1\) (of course \(x{\gt}0\) would suffice) and \(\sigma {\gt}0\), we have

For \(x{\gt}1\), we have

The rectangle (square) integral of \(f(s) = 1/s\) with corners \(-1-i\) and \(1+i\) is equal to \(2\pi i\).

Suppose that \(f\) is a holomorphic function on a rectangle, except for a simple pole at \(p\). By the latter, we mean that there is a function \(g\) holomorphic on the rectangle such that, \(f = g + A/(s-p)\) for some \(A\in \mathbb {C}\). Then the integral of \(f\) over the rectangle is \(A\).

We say that the Riemann hypothesis has been verified up to height \(T\) if there are no zeroes in the rectangle \(\{ \Re \rho \in (0.5, 1), \Im \rho \in [0,T] \} \).

An estimate of the form \(N(T) - \frac{T}{2\pi } \log \frac{T}{2\pi e} + \frac{7}{8}| \leq b_1 \log T + b_2 \log \log T + b_3\) for \(T \geq 2\).

For any natural \(N\ge 1\), we define

Conjugation symmetry of the Riemann zeta function. Let \(s \in \mathbb {C}\). Then

The log derivative of the Riemann zeta function \(\zeta (s)\) has a simple pole at \(s=1\) with residue \(-1\): \(-\frac{\zeta '(s)}{\zeta (s)} - \frac{1}{s-1} = O(1)\).

The Riemann zeta function \(\zeta (s)\) has a simple pole at \(s=1\) with residue \(1\). In particular, the function \(\zeta (s) - \frac{1}{s-1}\) is bounded in a neighborhood of \(s=1\).

\(\sum _{p \leq x} f(p) = \frac{f(x) \vartheta (x)}{\log x} - \int _2^x \vartheta (y) \frac{d}{dy}( \frac{f(y)}{\log y} )\ dy.\)

\(\vartheta (x) = x + O( x / \log ^2 x)\).

\(\sum _{p \leq x} f(p) = \int _{2}^x \frac{f(y)}{\log y}\ d\vartheta (y)\).

We have \(\psi (x) {\lt} c_0 x\) for all \(x{\gt}0\).

One has a Riemann von Mangoldt estimate with parameters 0.137, 0.443, and 1.588.

Given a complex function \(f\) and a real number \(M\), we define the function

where \(g\) is defined as in Definition 4.3.1.

The previous corollary also holds for functions \(\psi \) that are assumed to be in the Schwartz class, as opposed to being \(C^2\) and compactly supported.

The score of a factorization (relative to a cutoff parameter \(L\)) is equal to its waste, plus \(\log p\) for every surplus prime \(p\), \(\log (n/p)\) for every deficit prime above \(L\), \(\log L\) for every deficit prime below \(L\) and an additional \(\log n\) if one is not in total balance.

If one is in total balance, then the score is equal to the waste.

If there is a prime \(p\) in surplus, one can remove it without increasing the score.

If there is a prime \(p\) in deficit larger than \(L\), one can remove it without increasing the score.

If there is a prime \(p\) in deficit less than \(L\), one can remove it without increasing the score.

One can bring any factorization into balance without increasing the score.

If \(\psi : \mathbb {R}\to \mathbb {C}\) is absolutely integrable and \(x {\gt} 0\), then for any \(\sigma {\gt}1\)

Let \(R{\gt}0\) and \(f:\overline{\mathbb {D}_R}\to \mathbb {C}\). Define the set of zeros \(\mathcal{K}_f(R)=\{ \rho \in \mathbb {C}:|\rho |\leq R,\, f(\rho )=0\} \).

There exists \(C{\gt}0\) such that for all \(\delta \in (0,1)\) and \(t\in \mathbb {R}\) with \(|t|\geq 3\); if \(\zeta (\rho )=0\) with \(\rho =\sigma +it\), then

For all \(\delta \in (0,1)\) and \(t\in \mathbb {R}\) with \(|t|\geq 2\) we have

For all \(\delta \in (0,1)\) we have

\(\sigma (n)\) is the sum of the divisors of \(n\).

If \(I\) is a closed interval contained in an open interval \(J\), then there exists a smooth function \(\Psi : \mathbb {R}\to \mathbb {R}\) with \(1_I \leq \Psi \leq 1_J\).

Let \(\epsilon {\gt}0\). Then we define the smooth function \(\widetilde{1_{\epsilon }}\) from \(\mathbb {R}_{{\gt}0}\) to \(\mathbb {C}\) by

Fix a nonnegative, continuously differentiable function \(F\) on \(\mathbb {R}\) with support in \([1/2,2]\). Then for any \(\epsilon {\gt}0\), the function \(x \mapsto \int _{(0,\infty )} x^{1+it} \widetilde{1_{\epsilon }}(x) dx\) is continuous at any \(y{\gt}0\).

If \(\nu \) is nonnegative and has mass one, then \(\widetilde{1_{\epsilon }}(x)\le 1\), \(\forall x{\gt}0\).

If \(\nu \) is nonnegative, then \(\widetilde{1_{\epsilon }}(x)\) is nonnegative.

Fix \(0{\lt}\epsilon {\lt}1\). There is an absolute constant \(c{\gt}0\) so that: if \(x\geq (1+c\epsilon )\), then

Fix \(\epsilon {\gt}0\). There is an absolute constant \(c{\gt}0\) so that: If \(0 {\lt} x \leq (1-c\epsilon )\), then

Fix \(\epsilon {\gt}0\), and a bumpfunction supported in \([1/2,2]\). Then we define the smoothed Chebyshev function \(\psi _{\epsilon }\) from \(\mathbb {R}_{{\gt}0}\) to \(\mathbb {C}\) by

where we’ll take \(\sigma = 1 + 1 / \log X\).

We have that

We have that

Fix a nonnegative, continuously differentiable function \(F\) on \(\mathbb {R}\) with support in \([1/2,2]\), and total mass one, \(\int _{(0,\infty )} F(x)/x dx = 1\). Then for any \(\epsilon {\gt}0\), and \(\sigma \in (1, 2]\), the function

is integrable on \(\mathbb {R}\).

Fix a nonnegative, continuously differentiable function \(F\) on \(\mathbb {R}\) with support in \([1/2,2]\), and total mass one, \(\int _{(0,\infty )} F(x)/x dx = 1\). Then for any \(\epsilon {\gt}0\) and \(\sigma \in (1,2]\), the function \(x \mapsto \sum _{n=1}^\infty \frac{\Lambda (n)}{n^{\sigma +it}} \mathcal{M}(\widetilde{1_{\epsilon }})(\sigma +it) x^{\sigma +it}\) is equal to \(\sum _{n=1}^\infty \int _{(0,\infty )} \frac{\Lambda (n)}{n^{\sigma +it}} \mathcal{M}(\widetilde{1_{\epsilon }})(\sigma +it) x^{\sigma +it}\).

We have that

The integrand

is integrable on the contour \(\sigma _0 + t i\) for \(t \in \mathbb {R}\) and \(\sigma _0 {\gt} 1\).

We have that

We have that

There exists a smooth (once differentiable would be enough), nonnegative “bumpfunction” \(\nu \), supported in \([1/2,2]\) with total mass one:

We have

Let \(a {\lt} b\), and let \(\phi \) be continuously differentiable on \([a, b]\). Then

If ‘g‘ is a multiplicative arithmetic function, then for any \(n \neq 0\), \(\sum _{d | n} \mu (d) \cdot g(d) = \prod _{p \in n.\text{primeFactors}} (1 - g(p))\).

For all \(\delta \in (0,1)\) and \(t\in \mathbb {R}\) with \(|t|\geq 2\) we have

For all \(t\in \mathbb {R}\) with \(|t|\geq 2\) and \(z=\sigma +it\) where \(1-\delta _t/3\leq \sigma \leq 3/2\), we have that

For \(0 {\lt} t \leq 1/2\), one has \(| \frac{1}{\log (1+t)} + \frac{1}{\log (1-t)}| \leq \frac{16\log (4/3)}{3}\).